Evolution of quantum observables: from non-commutativity to commutativity

Abstract

A fundamental aspect of the quantum-to-classical limit is the transition from a non-commutative algebra of observables to commutative one. However, this transition is not possible if we only consider unitary evolutions. One way to describe this transition is to consider the Gamow vectors, which introduce exponential decays in the evolution. In this paper, we give two mathematical models in which this transition happens in the infinite time limit. In the first one, we consider operators acting on the space of the Gamow vectors, which represent quantum resonances. In the second one, we use an algebraic formalism from scattering theory. We construct a non-commuting algebra which commutes in the infinite time limit.

This is a preview of subscription content, log in to check access.

References

  1. Antoine JP (1969) Dirac formalism and symmetry problems in quantum mechanics. I. General Dirac formalism. J Math Phys 10:53–69

    MathSciNet  MATH  Google Scholar 

  2. Antoniou I, Laura R, Suchanecki Z, Tasaki S (1997) Intrinsic irreversibility of quantum systems with diagonal singularity. Phys A 241:737–772

    Google Scholar 

  3. Antoniou IE, Gadella M, Pronko GP (1998) Gamow vectors for degenerate scattering resonances. J Math Phys 39:2459–2475

    MathSciNet  MATH  Google Scholar 

  4. Antoniou I, Gadella M, Suchanecki Z (1998) Some general properties of Liouville spaces. In: Bohm A, Doebner HD, Kielanowski P (eds) Irreversibility and causality. Lecture notes in physics, vol 504. Springer, Berlin, pp 38–56

    Google Scholar 

  5. Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–843

    MathSciNet  MATH  Google Scholar 

  6. Bleistein N, Handelsman R (1986) Asymptotic expansion of integrals. Dover Inc., New York

    Google Scholar 

  7. Bohm A (1978) The rigged Hilbert space and quantum mechanics. Springer lecture notes in physics, vol 78. Springer, New York

    Google Scholar 

  8. Bohm A (1981) Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics. J Math Phys 22:2813–2823

    MathSciNet  Google Scholar 

  9. Bohm A (1993) Quantum mechanics: foundations and applications. Springer, Berlin

    Google Scholar 

  10. Bohm A, Gadella M (1989) Dirac kets, Gamow vectors and Gelfand triplets. Springer lecture notes in physics, vol 348. Springer, New York

    Google Scholar 

  11. Bohm A, Erman F, Uncu H (2011) Resonance phenomena and time asymmetric quantum mechanics. Turk J Phys 35:209–240

    Google Scholar 

  12. Castagnino M, Fortin S (2013) Formal features of a general theoretical framework for decoherence in open and closed systems. Int J Theor Phys 52:1379–1398

    MathSciNet  MATH  Google Scholar 

  13. Castagnino M, Gadella M (2006) The problem of the classical limit of quantum mechanics and the role of self-induced decoherence. Found Phys 36:920–952

    MathSciNet  MATH  Google Scholar 

  14. Castagnino M, Lombardi O (2005) Self-induced decoherence and the classical limit of quantum mechanics. Philos Sci 72:764–776

    MathSciNet  Google Scholar 

  15. Castagnino M, Gadella M, Gaioli F, Laura R (1999) Gamow vectors and time asymmetry. Int J Theor Phys 38:2823–2865

    MathSciNet  MATH  Google Scholar 

  16. Castagnino M, Gadella M, Betán RI, Laura R (2001) Gamow functionals on operator algebras. J Phys A Math Gen 34:10067–10083

    MathSciNet  MATH  Google Scholar 

  17. Castagnino M, Fortin S, Lombardi O (2010) The effect of random coupling coefficients on decoherence. Mod Phys Lett A 25:611–617

    MATH  Google Scholar 

  18. Celeghini E, Gadella M, del Olmo MA (2016) Applications of rigged Hilbert spaces in quantum mechanics and signal processing. J Math Phys 57:072105

    MathSciNet  MATH  Google Scholar 

  19. Celeghini E, Gadella M, del Olmo MA (2017) Lie algebra representations and rigged Hilbert spaces: the SO(2) case. Acta Polytech (Prag) 57:379–384

    Google Scholar 

  20. Celeghini E, Gadella M, del Olmo MA (2018) Spherical harmonics and rigged Hilbert spaces. J Math Phys 59:053502

    MathSciNet  MATH  Google Scholar 

  21. Civitarese O, Gadella M (2004) Physical and mathematical aspects of Gamow states. Phys Rep 396:41–113

    MathSciNet  Google Scholar 

  22. Dalla Chiara ML, Giuntini R, Greechie R (2004) Reasoning in quantum theory. Kluwer Academic, Dordrecht

    Google Scholar 

  23. Exner P (1984) Open quantum systems and Feynman integrals. Reidel, Dordrecht

    Google Scholar 

  24. Fischer MC, Gutiérrez-Medina B, Reizen MG (2001) Observation of the quantum Zeno and anti-Zeno effects in an unstable system. Phys Rev Lett 87:40402

    Google Scholar 

  25. Fonda L, Ghirardi GC, Rimini A (1978) Decay theory of unstable quantum systems. Rep Prog Phys 41:587–631

    Google Scholar 

  26. Fortin S, Vanni L (2014) Quantum decoherence: a logical perspective. Found Phys 44:1258–1268

    MathSciNet  MATH  Google Scholar 

  27. Fortin S, Holik F, Vanni L (2016) Non-unitary evolution of quantum logics. In: Bagarello F, Passante R, Trapani C (eds) Non-hermitian Hamiltonians in quantum physics. Springer proceedings in physics, vol 184. Springer, Cham

    Google Scholar 

  28. Friedrichs KO (1948) On the perturbation of continuous spectra. Commun Appl Math 1:361–406

    MathSciNet  MATH  Google Scholar 

  29. Gadella M (2014) Quantum resonances: theory and models. In: Kielanowski P, Bieliavsky P, Odesskii A, Odzijewicz A, Schlichenmaier M, Voronov T (eds) Geometric methods in physics, XXXII workshop Bialowieza. Springer Basel AG, Poland, pp 99–118

  30. Gadella M (2015) A discussion on the properties of Gamow states. Found Phys 45:177–197

    MathSciNet  MATH  Google Scholar 

  31. Gadella M, Gómez F (2002) A unified mathematical formalism for the Dirac formulation of quantum mechanics. Found Phys 32:815–869

    MathSciNet  Google Scholar 

  32. Gadella M, Gómez F (2003) On the mathematical basis of the Dirac formulation of quantum mechanics. Int J Theor Phys 42:2225–2254

    MathSciNet  MATH  Google Scholar 

  33. Gadella M, Laura R (2001) Gamow dyads and expectation values. Int J Quantum Chem 81:307–320

    Google Scholar 

  34. Gadella M, de la Madrid R (1999) Resonances and time reversal operator in rigged Hilbert spaces. Int J Theor Phys 38:93–113

    MathSciNet  MATH  Google Scholar 

  35. Gadella M, Pronko GP (2011) The Friedrichs model and its use in resonance phenomena. Fortschr Phys 59:795–859

    MathSciNet  MATH  Google Scholar 

  36. Gadella M, Kuru Ş, Negro J (2017) The hyperbolic step potential: antibound states, SUSY partners and Wigner time delays. Ann Phys 379:86–101

    MATH  Google Scholar 

  37. Gelfand IM, Vilenkin NY (1964) Generalized functions: applications to harmonic analysis. Academic, New York

    Google Scholar 

  38. Gell-Mann M, Hartle JB (1990) Quantum mechanics in the light of quantum cosmology. In: Zurek WH (ed) Complexity, entropy and the physics of information. Addison-Wesley, Reading

    Google Scholar 

  39. Gell-Mann M, Hartle JB (1993) Classical equations for quantum systems. Phys Rev D 47:3345–3382

    MathSciNet  Google Scholar 

  40. Griffiths RB (2002) Consistent quantum theory. Cambridge University Press, Cambridge

    Google Scholar 

  41. Horvath J (1966) Topological vector spaces and distributions. Addison-Wesley, Reading

    Google Scholar 

  42. Khalfin LA (1972) CPT invariance of CP-noninvariant theory of K0 and Kbar0 Mesons and permissible mass distributions of the KS and KL Mesons. JETP Lett 15:388–392

    Google Scholar 

  43. Kiefer C, Polarski D (2009) Why do cosmological perturbations look classical to us? Adv Sci Lett 2:164–173

    Google Scholar 

  44. Losada M, Fortin S, Holik F (2018) Classical limit and quantum logic. Int J Theor Phys 57:465–475

    MathSciNet  MATH  Google Scholar 

  45. Losada M, Fortin S, Gadella M, Holik F (2018) Dynamics of algebras in quantum unstable systems. Int J Mod Phys A 33:1850109

    MathSciNet  MATH  Google Scholar 

  46. Melsheimer O (1974) Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory. J Math Phys 15:902–916

    MathSciNet  Google Scholar 

  47. Misra B, Sudarshan ECG (1977) The Zeno’s paradox in quantum theory. J Math Phys 18:756–763

    MathSciNet  Google Scholar 

  48. Mondragón A, Hernández E (1993) Degeneracy and crossing of resonance energy surfaces. J Phys A Math Gen 26:5595–5611

    MathSciNet  Google Scholar 

  49. Nakanishi N (1958) A theory of clothed unstable particles. Progr Theor Phys 19:607–621

    MathSciNet  MATH  Google Scholar 

  50. Nussenzveig HM (1972) Causality and dispersion relations. Academic Press, New York

    Google Scholar 

  51. Omnès R (1999) Understanding quantum mechanics. Princeton University Press, Princeton

    Google Scholar 

  52. Ramírez R, Reboiro M (2019) Dynamics of finite dimensional non-hermitian systems with indefinite metric. J Math Phys 60:012106

    MathSciNet  MATH  Google Scholar 

  53. Ramírez R, Reboiro M (2019) Optimal spin squeezed steady state induced by the dynamics of non-hermitian Hamiltonians. Phys Scr 94:085220

    Google Scholar 

  54. Reed M, Simon B (1978) Analysis of operators. Academic Press, New York

    Google Scholar 

  55. Reed M, Simon B (1981) Functional analysis. Academic Press, New York

    Google Scholar 

  56. Roberts JE (1966) Rigged Hilbert spaces in quantum mechanics. Commun Math Phys 3:98–119

    MathSciNet  MATH  Google Scholar 

  57. Rothe C, Hintschich SI, Monkman AP (2006) Violation of the exponential-decay law at long times. Phys Rev Lett 96:163601

    Google Scholar 

  58. Schlosshauer M (2007) Decoherence and the quantum-to-classical transition. Springer, Berlin

    Google Scholar 

  59. Urbanowski K (2009) General properties of the evolution of unstable states at long times. Eur Phys J D 54:25–29

    Google Scholar 

  60. Wigner EP (1967) Symmetries and reflections. Indiana University Press, Bloomington, pp 38–39

    Google Scholar 

  61. Wigner EP (1994) Group theoretical concepts and methods in elementary particle physics. Gordon and Breach, New York, pp 37–38

    Google Scholar 

  62. Zurek WH (2009) Quantum darwinism. Nat Phys 5:181–188

    Google Scholar 

Download references

Acknowledgements

M. Gadella acknowledges partial financial support to the Spanish Government Grant MTM2014-57129-C2-1-P, the Junta de Castilla y León Grants BU229P18, VA137G18. S. Fortin, F. Holik and M. Losada wish to acknowledge the financial support of the Universidad de Buenos Aires, the Grant PICT-2014-2812 from the Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina.

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. Losada.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by F. Holik.

Appendix

Appendix

Let us show that formulas (41) and (42) are not equivalent. For simplicity, we shall assume that there is only one resonance so that dim \({\mathcal {H}}^G=2\). The vector \(|\psi )\) being arbitrary is a linear combination of \(|\psi ^D\rangle \) and \(|\psi ^G\rangle \), so that it may be written as a column vector as \(\left( \begin{array}{l} a\\ b\\ \end{array} \right) \) with a and b complex. Then,

$$\begin{aligned} (\psi ^D|\psi )= & {} (1,0) B^\dagger \, \left( \begin{array}{l} a \\ b\end{array}\right) \nonumber \\= & {} (1,0) \,i^{1/2} \left( \begin{array}{ll} -i{\sqrt{2}}/2 &{}\quad {\sqrt{2}}/2 \\ {\sqrt{2}}/2 &{}\quad -i{\sqrt{2}}/2\end{array}\right) \left( \begin{array}{l} a \\ b\end{array}\right) \nonumber \\= & {} i^{1/2} \left[ \frac{{\sqrt{2}}}{2}\,b -i\,\frac{{\sqrt{2}}}{2}\,a\right] . \end{aligned}$$
(63)

Similar calculations yield (we have written \(\sqrt{-1}=-i\)):

$$\begin{aligned} (\psi ^G|\psi )= & {} (\psi ^D|\psi ),\nonumber \\ (\psi |\psi ^D)= & {} (\psi |\psi ^G)=i^{1/2} \left[ \frac{{\sqrt{2}}}{2}\,b^* -i\,\frac{{\sqrt{2}}}{2}\,a^* \right] . \end{aligned}$$
(64)

This gives (41). To obtain (42), we need the following calculation:

$$\begin{aligned} (\psi ^D|T|\psi )= & {} (1,0) \, i^{1/2} \left( \begin{array}{ll} -i{\sqrt{2}}/2 &{}\quad {\sqrt{2}}/2 \\ {\sqrt{2}}/2 &{}\quad -i{\sqrt{2}}/2\end{array}\right) \left( \begin{array}{ll} 0 &{}\quad C \\ C &{}\quad 0 \end{array}\right) \left( \begin{array}{l} a \\ b\end{array}\right) \nonumber \\= & {} i^{1/2}\, \left( -i\,{\sqrt{2}}/2, {\sqrt{2}}/2 \right) \left( \begin{array}{l} b^* \\ a^*\end{array}\right) \nonumber \\= & {} i^{1/2}\, \left[ \frac{{\sqrt{2}}}{2}\,a^* -i\, \frac{{\sqrt{2}}}{2}\,b^* \right] . \end{aligned}$$
(65)

Taken \(\sqrt{-1}=-i\), we easily find that \((\psi ^D|T|\psi )=(\psi ^G|T|\psi )\). From the other terms, we write

$$\begin{aligned} (\psi |T|\psi ^D)= & {} (a^*,b^*) \left( \begin{array}{ll} 0 &{}\quad C \\ C &{}\quad 0 \end{array}\right) (-i)^{1/2} \left( \begin{array}{ll} i{\sqrt{2}}/2 &{}\quad {\sqrt{2}}/2 \\ {\sqrt{2}}/2 &{}\quad i{\sqrt{2}}/2\end{array}\right) \left( \begin{array}{l} 1 \\ 0\end{array}\right) \nonumber \\= & {} i^{1/2} (a^*,b^*) \left( \begin{array}{ll} 0 &{}\quad C \\ C &{}\quad 0 \end{array}\right) \left( \begin{array}{l}{\sqrt{2}}/2 \\ -i{\sqrt{2}}/2 \end{array}\right) \nonumber \\= & {} i^{1/2}\, \left[ \frac{{\sqrt{2}}}{2}\,a^* -i\, \frac{{\sqrt{2}}}{2}\,b^* \right] . \end{aligned}$$
(66)

Similarly, we obtain that

$$\begin{aligned} (\psi |T|\psi ^G)=(\psi ^G|T|\psi )=(\psi ^D|T|\psi )=(\psi |T|\psi ^D). \end{aligned}$$
(67)

The obvious conclusion is that (41) and (42) do not coincide. The first and third identities in (67) are not a surprise due to the properties of the time reversal operator T.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fortin, S., Gadella, M., Holik, F. et al. Evolution of quantum observables: from non-commutativity to commutativity. Soft Comput 24, 10265–10276 (2020). https://doi.org/10.1007/s00500-019-04546-7

Download citation

Keywords

  • Foundations of quantum mechanics
  • Gamow vectors
  • Rigged Hilbert space
  • Classical limit
  • Quantum resonances